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APT assumes that idiosyncratic risk is zero on average: $E[e_i]=0$. The law of large numbers. From 1 and 2 it follows that as N increases, the weighted sum of idiosyncratic risks will converge to zero: $\lim\limits_{N\to\infty}\sum\limits_{i=1}^N e_p=\lim\limits_{N\to\infty}\sum\limits_{i=1}^N w_ie_i=0$ Strictly speaking some restrictions on the weights ...


$\alpha$ units of cash and $\beta$ bonds? Presumably you mean 'value' rather than 'return', since the SDF is not a percentage return but a 'discount factor'.

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